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Title Finite morphisms onto Fano manifolds of Picard number 1 whichhave rational curves with trivial normal bundles

Author(s) Hwang, JM; Mok, N

Citation Journal Of Algebraic Geometry, 2003, v. 12 n. 4, p. 627-651

Issued Date 2003

URL http://hdl.handle.net/10722/42125

Rights Creative Commons: Attribution 3.0 Hong Kong License

J. ALGEBRAIC GEOMETRY12 (2003) 627–651S 1056-3911(03)00319-9Article electronically published on April 10, 2003

FINITE MORPHISMS ONTO FANO MANIFOLDSOF PICARD NUMBER 1

WHICH HAVE RATIONAL CURVESWITH TRIVIAL NORMAL BUNDLES

JUN-MUK HWANG AND NGAIMING MOK

Abstract

Let X be a Fano manifold of Picard number 1 admitting a rational curvewith trivial normal bundle and f : X′ → X be a generically finite sur-jective holomorphic map from a projective manifold X′ onto X. Whenthe domain manifold X′ is fixed and the target manifold X is a pri-ori allowed to deform we prove that the holomorphic map f : X′ → Xis locally rigid up to biholomorphisms of target manifolds. This resultcomplements, with a completely different method of proof, an earlierlocal rigidity theorem of ours (see J. Math. Pures Appl. 80 (2001), 563–575) for the analogous situation where the target manifold X is a Fanomanifold of Picard number 1 on which there is no rational curve withtrivial normal bundle. In another direction, given a Fano manifold X′

of Picard number 1, we prove a finiteness result for generically finitesurjective holomorphic maps of X′ onto Fano manifolds (necessarily ofPicard number 1) admitting rational curves with trivial normal bundles.As a consequence, any 3-dimensional Fano manifold of Picard number 1can only dominate a finite number of isomorphism classes of projectivemanifolds.

There are various results concerning finiteness or rigidity of dominant mor-phisms or rational maps from a fixed projective variety onto varieties of non-positive curvature. For example, Kobayashi-Ochiai showed ([KO]) that forany algebraic variety Y and any algebraic variety Z of general type, the num-ber of dominant rational maps from Y to Z is finite. Another example isMaehara’s result ([Ma]) that for a given projective algebraic manifold Y , thenumber of minimal projective manifolds of general type, which can be theimage of a dominant rational map from Y , is finite. Eckart Viehweg hadinformed us that the works on the semi-positivity of direct images of tensorpowers of dualizing sheaves, due to Kawamata, Viehweg and others, implythat for any projective algebraic variety Y , there are only countably many

Received December 18, 2000. Supported by Grant No. 98-0701-01-5-L from the KOSEF.Supported by a grant of the Hong Kong Research Grants Council.

627

628 JUN-MUK HWANG AND NGAIMING MOK

minimal (not necessarily of general type) projective manifolds which can bethe image of Y under a morphism.

On the other hand, there are very few results of this type when the targetshave positive curvature, i.e. when the targets are Fano manifolds. In [HM1],the authors showed that a projective manifold, which can be the image of arational homogeneous space G/P of Picard number 1 under a holomorphicmap, must be either G/P itself or the projective space. As a natural gen-eralization of this result, one may ask which projective manifolds (which areautomatically Fano) can be the image of a holomorphic map from a fixed Fanomanifold Y of Picard number 1. However, for a general Y , the possible imagescan be more than just itself or the projective space, as one can easily see inthe case of some Fano hypersurfaces in the projective space. So it seems thata more modest question about the finiteness of such images is already worthstudying:

Question. Given a Fano manifold Y of Picard number 1, are there onlya finite number of isomorphism classes of projective manifolds which can bethe image of Y under a holomorphic map?

One of the results of this paper is an affirmative answer to the Question indimension 3. Our main results hold in any dimension, but we need to imposean additional condition on the image manifolds. Let us explain this condition.

Let X be a Fano manifold of Picard number 1. It is easy to see from thedeformation theory of rational curves (e.g. [Kl]) that the following conditionson X are equivalent.

(1) There exists a rational curve µ : P1 → X so that µ∗T (X) ∼= O(2) ⊕On−1.

(2) For a generic point x ∈ X , there are only finitely many rational curvesthrough x which have minimal degree with respect to K−1

X .(3) For a generic point x ∈ X , there exists a rational curve which has

degree 2 with respect to K−1X .

A rational curve satisfying the first condition will be called a rational curvewith trivial normal bundle. The rational curve in the second and the thirdconditions is a rational curve with trivial normal bundle.

There are many examples of Fano manifolds which satisfy the above equiv-alent conditions. When n = 3, excepting the projective space P3 and the3-dimensional hyperquadric, all Fano 3-folds of Picard number 1 satisfy thecondition ([Is]). Also, for any m ≥ 4, a smooth hypersurface of degree m− 1or m in Pm satisfies it. When the degree is m−1, there are finitely many linesthrough a generic point. When the degree is m, there exist no lines througha generic point, but finitely many conics through a generic point.

FINITE MORPHISMS ONTO FANO MANIFOLDS OF PICARD NUMBER 1 629

In this paper, we will study finite morphisms over Fano manifolds of Picardnumber 1 which have rational curves with trivial normal bundles. We havetwo main results. The first one is the deformation rigidity of such morphismsin the following sense:

Theorem 1. Let χ : X → ∆ := t ∈ C, |t| < 1 be a regular family ofFano manifolds of Picard number 1 so that X0 = χ−1(0) has rational curveswith trivial normal bundles. For a given projective manifold X ′, suppose thereexists a surjective morphism f : X ′ = X ′ ×∆→ X respecting the projectionsto ∆ so that ft : X ′ → Xt is a generically finite morphism for each t ∈ ∆.Then there exists ε > 0 and a holomorphic family of biholomorphic morphismsvt : Xo → Xt, |t| < ε, satisfying ft = vt fo, v0 = id.

In particular, the set Hol(X ′, X) of generically finite morphisms from acomplete variety X ′ onto X is countable up to automorphisms of X . Ananalogous result was obtained in [HM3] for a class of Fano manifolds of Picardnumber 1 which do not have rational curves with trivial normal bundles. Themethods employed in [HM3] cannot be applied to the current case at all andhave little to do with the techniques used in this paper.

Our second result is the boundedness of the degrees of the finite morphismswhen the domain X ′ is also a Fano manifold of Picard number 1.

Theorem 2. Let X and X ′ be n-dimensional Fano manifolds of Picardnumber 1. Assume that X has rational curves with trivial normal bundles.Then there exists a positive number N determined by X ′ such that for anyfinite morphism f : X ′ → X, the degree of f is bounded by N .

As a consequence of Theorems 1 and 2, we obtain the following partialanswer to the Question.

Corollary 4. Let X ′ be a Fano manifold of Picard number 1. Amid Fanomanifolds of Picard number 1 which have rational curves with trivial normalbundles, only finitely many can be the image of a holomorphic map from X ′.Furthermore, for each such X there are at most a finite number of non-trivialholomorphic maps f : X ′ → X up to automorphisms of X.

As mentioned above, among Fano threefolds only the hyperquadric andthe projective space do not have rational curves with trivial normal bundles.Thus, Corollary 4 gives an affirmative answer to the Question in dimension 3.

It should be mentioned that Theorem 2 was proved in dimension 3 by [Am]and [ARV]. In fact, they proved the same result for any (not necessarily Fano)projective threefold X ′ of Picard number 1. In dimension 3, many additionalresults on morphisms to Fano threefolds were obtained in [Am], [ARV], [IS]and [Sc], many of which cannot be covered by our results. On the other hand,their methods rely heavily on the classification of Fano threefolds while ours isbasically independent of the classification theory. Moreover, their methods do


not seem applicable to the question of local rigidity (Theorem 1). For example,our Corollary 2, that finite morphisms from Mukai-Umemura threefolds ontoFano threefolds different from P3 must be isomorphisms, is out of reach oftheir methods.

Our approach is based on the philosophy that many problems on Fanomanifolds can be solved by looking at tangent directions to minimal rationalcurves (see [HM1], [HM2] for other examples of this philosophy). The tan-gent directions to rational curves with trivial normal bundles form a finitebranched covering of the Fano manifold. The main idea of the paper is tostudy the geometry of the branch locus of this covering and examine how itchanges under a finite morphism. The crucial point in applying this study toproblems on generically finite morphisms is that the inverse image of rationalcurves with trivial normal bundles under a generically finite morphism alsohave trivial normal bundles (Proposition 6). In general, these inverse imagesare not rational curves. So it is necessary to study curves with trivial normalbundles of arbitrary genus. Compared with rational curves, the deformationtheory of curves of high genus could be quite tricky. However, in the situationwe are studying, no obstructions appear and the deformation theory is quiteparallel to that of rational curves with trivial normal bundles. The first twosections present a general theory of curves with trivial normal bundles cov-ering a projective manifold. The next two sections study its behavior undergenerically finite morphisms. The proof of Theorem 1, which will be given inSection 5, uses only Section 1 and Section 3 and is independent of Section 2and Section 4. The proof of Theorem 2 will be given in Section 6.

Convention and terminology

1. All varieties and morphisms are defined over complex numbers. Opensets of a variety mean Euclidean open sets, not Zariski open sets. A varietyneed not be irreducible, but has finitely many components. A generic point ofa variety of pure dimension means a generic point of any of its components. Sowhen we say that a certain statement holds for a generic point of a variety ofpure dimension, it means that it holds for a generic point of each componentof the variety.

2. A projective manifold means an irreducible smooth projective variety. AFano manifold means a projective manifold with ample anti-canonical bundle.

3. For a proper morphism between two varieties ϕ : Y → X , the inverseimage ϕ−1(Z) of a subvariety Z ⊂ X is taken in the set-theoretical sense andunderstood as a reduced subvariety of Y .


4. Suppose a generically finite morphism ϕ : Y → X , from a completevariety Y to a projective manifold X , is given. Consider the set

y ∈ Y, Y is singular at y or ϕ is ramified at y.

The ramification divisor of ϕ is the union of codimension-1 components ofthe above set which are not contracted to lower dimensional subvarieties of Xby ϕ. The branch divisor of ϕ is the image of the ramification divisor. LetB ⊂ X be the branch divisor. Then ϕ is unramified over X −B−Z for somesubvariety Z ⊂ X of codimension > 1. For a point y ∈ Y where ϕ is locallyfinite, the degree of the germ of ϕ at y will be called the local sheetingnumber of ϕ at y. Given a point x ∈ X , an isolating neighborhood of xwith respect to ϕ means a connected open subset U ⊂ X containing x suchthat for each irreducible component G of ϕ−1(U), G∩π−1(x) is a single point.There exists a subvariety of codimension > 1 in X so that any point outsidethis subvariety has an isolating neighborhood.

1. Webs and their discriminantal divisors

Let X be a projective manifold of dimension n and C ⊂ X be an irreduciblereduced curve. We say that C has a trivial normal bundle if under thenormalization µ : C → C, we get the following exact sequence

0 −→ ΘC −→ µ∗ΘX −→ On−1

C−→ 0

where Θ denotes the tangent sheaf. In this case, µ : C → X is an immersion.Suppose that C has a trivial normal bundle and deformations of C withconstant geometric genus cover an open subset of X . The germ of the spaceof deformations of C with constant geometric genus must have dimension≥ n− 1. The Zariski tangent space to this space at the point correspondingto C is H0(C, µ∗ΘX/ΘC) (cf. [HM1], p. 212), which has dimension n−1 fromthe triviality of the normal bundle. Thus the germ is smooth. The closureMC of this germ in the Hilbert scheme of X will be called the irreducibleweb defined by C. The underlying variety ofMC is irreducible, projective, ofdimension n−1 and smooth at the point [C] corresponding to C. Conversely,an irreducible subscheme of dimension n−1 in the Hilbert scheme of curves onX is called an irreducible web if its members cover X and generic membershave trivial normal bundles. A web is a finite collection of irreducible webs.There are only finitely many irreducible webs in an irreducible component ofthe Hilbert scheme of curves on X . Thus there are only countably many websfor a given X .


For a web M, let ψ : F → M and φ : F → X be the universal familymorphisms. We will assume that F and M are normal projective varietiesby taking the normalization of the underlying reduced structures of all theschemes involved. Then ψ need not be flat any more, but every fiber of ψ is ofpure dimension 1 and a generic fiber of ψ is an irreducible smooth curve. Notethat F andM are varieties of pure dimension. Clearly, φ is generically finite.If φ is d-to-1, d is called the degree of the web M. In a neighborhood of ageneric point of X , the images of fibers of ψ define d distinct local foliationsby open pieces of curves.

Remark. In local differential geometry, a finite collection of foliations ofequal rank is called a web. At a generic point of X a web in our sense definesa web of rank-1 in the differential-geometric sense.

From the triviality of the normal bundle, φ is unramified in a neighborhoodof a generic fiber of ψ. It follows that for a given subvariety Z ⊂ X ofcodimension > 1, a generic member of M is disjoint from Z.

We say that a connected curve C in a smooth variety is transversal toa hypersurface H if either C ∩ H is empty or it consists of finitely manysmooth points of H where each local irreducible germ of C is smooth and isnot tangent to H .

Proposition 1. Let M be a web on X. Let H ⊂ X be an irreduciblehypersurface and y ∈ H be a generic point. Then for any v ∈ M with y ∈φ(ψ−1(v)), one of the following holds.

(i) φ(ψ−1(v)) has trivial normal bundle and is transversal to H.(ii) φ(ψ−1(v)) is locally irreducible at y and its component containing y

is contained in H.

Proof. Since φ is unramified in a neighborhood of a generic fiber of ψ, thestatement (i) holds for a generic v ∈M. Let

N := v ∈ M, (i) does not hold for v.

Then dim(N ) ≤ n − 2 and φ(ψ−1(N )) is a proper subvariety of X . WhenH 6⊂ φ(ψ−1(N )), choose y ∈ H − φ(ψ−1(N )). Then for any v ∈ M with y ∈φ(ψ−1(v)), (i) is always satisfied. When H ⊂ φ(ψ−1(N )), let φ(ψ−1(N )) =H∪G where G consists of components different from H and choose y ∈ H−G.Then for any v ∈ M with y ∈ φ(ψ−1(v)), either Case (1) v 6∈ N and (i)holds, or Case (2) v ∈ N and φ(ψ−1(v)) ⊂ H ∪G. In Case (2), an irreduciblecomponent of φ(ψ−1(v)) containing y is contained inH . Since dim(N ) ≤ n−2,such a curve is locally irreducible at a generic point of H .

Let Φ : F → PT (X) be the rational map defined by

Φ(a) := PTφ(a)(φ(ψ−1(ψ(a))))


for a generic point a ∈ F . Namely, Φ(a) is the tangent direction of the curvecorresponding to ψ(a) ∈ M at the point φ(a). Φ will be called the tangentmap. Let C ⊂ PT (X) be the strict image of Φ. C will be called the varietyofM-tangents.

Proposition 2. Let π : C → X be the restriction of the projection PT (X)→ X to C. Then π is generically d-to-1, where d is the degree of the web.Moreover, there exists a subvariety E of codimension ≥ 2 in X so that Φis well-defined on F − φ−1(E) and can be regarded as the normalization ofC|X−E.

Proof. Suppose π : C → X is generically m-to-1 for m ≤ d. At a genericpoint x ∈ X where π is unramified, C ⊂ PT (X) defines m distinct localfoliations by open pieces of curves. This must agree with d distinct foliationsby open pieces of curves defined by the φ-images of the ψ-fibers. Thus m = d.It follows that Φ : F → C is birational. Since Φ respects the morphismsφ : F → X and π : C → X , the second statement is immediate.

The branch divisor of φ : F → X will be called the discriminantal divisorof the web M and denoted by DM. The branch divisor of π : C → X

will be called the extended discriminantal divisor of M and denotedby EM. From Proposition 2, it is clear that DM ⊂ EM. When y is asmooth point of DM, let PTy(DM) be the projectivized tangent space. LetPT (DM) ⊂ PT (X) be the subvariety defined as the closure of the union ofPTy(DM) as y varies over the smooth points of DM. The variety PT (EM)is defined similarly.

Proposition 3. Let R ⊂ F be the ramification divisor of φ : F → X.Then Φ(R) ⊂ PT (DM).

Proof. Recall that F and M are normal varieties. Since φ is unramifiedin a neighborhood of a generic fiber of ψ, a generic fiber of ψ is disjoint fromR and R consists of certain components of fibers of ψ over a hypersurfacein M. These components of fibers of ψ are sent to curves in DM = φ(R).Generic points of Φ(R) correspond to tangent vectors to these curves, henceΦ(R) ⊂ PT (DM) by Proposition 1.

2. Discriminantal order

LetM be a web on a projective manifold X . In this section, we will defineintegers δL for certain components L of the extended discriminantal divisorEM ⊂ X .

Let L be an irreducible component of EM and let y ∈ L be a generic point.Let U be an isolating neighborhood of y with respect to the generically finite


morphism π : C → X (cf. Convention and Terminology, 4). Let G be theunion of components of C|U which intersect PTy(L). When G is non-empty, itis generically m-to-1 over U for some positive integer m. This m is certainlyindependent of the choice of the generic point y and depends only on thechoice of the component L of EM. We will call it the tangential sheetingnumber of M along L and denote it by τL.

Proposition 4. If M is a component of DM, then τM > 1.Proof. Let R be a component of the ramification divisor of φ : F → X

over M . For a generic point y of M , any germ of R over y is smooth becauseF is normal. Since φ is ramified at y, the local sheeting number > 1. ByProposition 3, Φ(R) is contained in PT (M). Thus for a generic point y ∈M ,any germ of C at a generic point of Φ(R) gives a germ of G where the localsheeting number of π is strictly bigger than 1.

Suppose L is a component of EM with τL > 1. Let ξ be the tautologicalline bundle on the projectivized tangent bundle PT (X). By shrinking U ifnecessary, we may assume that ξ is trivial on G. Let V be a non-vanishingsection of ξ over G, the normalization of G. In terms of a coordinate system(z1, . . . , zn) centered at y ∈ L, V can be written as

V = v1∂

∂z1+ · · ·+ vn

∂

∂zn

where vi’s are multi-valued holomorphic functions on U , or more precisely,holomorphic functions on G. Set m = τL. For a generic point x ∈ U , letx1, . . . , xm be the points of G over x. As x approaches y, at least two ofx1, . . . , xm get closer from the definition of EM. For an antisymmetric n× ncomplex matrix Q = (qij) ∈ o(n,C), consider the holomorphic function on Udefined by

ΓQ(x) :=∏

1≤α6=β≤m

n∑i,j=1

qijvi(xα)vj(xβ)

for generic x ∈ U . Let γQ ≥ 1 be the vanishing order of ΓQ along L and set

δy := minQ∈o(n,C)

γQ.

Proposition 5. The positive integer δy is uniquely determined by thechoice of y ∈ L, namely, it is independent of the choice of a local coordi-nate system at y and the choice of the multi-valued vector field V .

Proof. First, we will show that once V is chosen, then δy is independentof the choice of coordinates. Let Γ]Q, γ

]Q and δ]y be as defined above using

another coordinate system (z]1, . . . , z]n) given by z]i = Ψi(z1, . . . , zn) for some


holomorphic functions Ψi, 1 ≤ i ≤ n. Let

V = v]1∂

∂z]1+ · · ·+ v]n

∂

∂z]n.

Then we have

v]i =n∑k=1

vk∂Ψi

∂zk.

It follows that

Γ]Q(x) :=∏

1≤α6=β≤m

n∑i,j=1

qijv]i (xα)v]j(x

β)

=

∏1≤α6=β≤m

n∑i,j,k,l=1

qijvk(xα)vl(xβ)∂Ψi

∂zk

∂Ψj

∂zl

=

∏1≤α6=β≤m

b∑k,l=1

qkl] vk(xα)vl(xβ)

= ΓQ](x)

where Q] is the antisymmetric matrix with entries

qkl] =n∑

i,j=1

qij∂Ψi

∂zk

∂Ψj

∂zl.

It follows that the vanishing order of Γ]Q along L is greater than or equal toδy. This shows that δy ≤ δ]y. Applying the same argument in reverse, we getδy ≥ δ]y. It follows that δy = δ]y.

Now let us show that δy is independent of the choice of the section V of ξon G. Let V [ be another non-vanishing section and let Γ[Q, δ

[y be as defined

above using V [. Then V [ = hV for some non-vanishing holomorphic functionh on G. With respect to the coordinate system (z1, . . . , zn),

V [ = hv1∂

∂z1+ · · ·+ hvn

∂

∂zn,

from which we get

Γ[Q(x) =∏

1≤α6=β≤m

n∑i,j=1

qijh(xα)vi(xα)h(xβ)vj(xβ)

= ΓQ(x) ·

∏1≤α6=β≤m

(h(xα)h(xβ)

).

Since∏

1≤α6=β≤m(h(xα)h(xβ)

)is non-vanishing on L, we see that δy = δ[y.


It is easy to see that δy does not depend on the choice of the generic pointy ∈ L. It will depend only on the choice of an irreducible component L ofEM, satisfying the condition τL > 1. We call it the discriminantal orderof L.

3. Inverse image webs

Let f : X ′ → X be a generically finite morphism between two projectivemanifolds. Let B ⊂ X be the branch divisor and R ⊂ X ′ be the ramificationdivisor of f . Let Kerdf ⊂ PT (X ′) be the projectivization of the kernel ofthe differential df : T (X ′) → T (X). Note that for a generic point z ∈ R,Kerdf ∩ PTz(R) = ∅. We denote the projectivization of df by the samesymbol df : PT (X ′) → PT (X). This is a rational map which is well-definedoutside Kerdf .

Proposition 6. Given a generically finite morphism f : X ′ → X betweentwo projective manifolds of dimension n, suppose there exists a web M for Xand let C ⊂ X be a generic member. Then each irreducible component C′ off−1(C) has trivial normal bundle.

We start with an elementary lemma.Lemma 1. Let f : X ′ → X be a generically finite morphism between two

compact complex manifolds. Let x ∈ B be a generic point of the branch divisorand ∆ ⊂ X be a germ of smooth complex analytic curve through x intersectingB transversally. Then f−1(∆) is also smooth.

Proof of Lemma 1. Choose a local coordinate system z1, . . . , zn centeredat x and a local coordinate system w1, . . . , wn centered at a point x′ ∈ f−1(x)such that f is given by

z1 = w1, . . . , zn−1 = wn−1, zn = wrn

where r is the local sheeting number and B is defined by zn = 0. Since ∆ istransversal to B, it is given by equations of the form

z1 = h1(zn), . . . , zn−1 = hn−1(zn)

for some convergent power series h1, . . . , hn−1 of one variable zn. Thusf−1(∆) is defined near x′ by

w1 = h1(wrn), . . . , wn−1 = hn−1(wrn),

which is smooth. Proof of Proposition 6. Let µ : C → C be the normalization. From the

genericity of C and Proposition 1, C intersects B transversally. Thus the


normalization µ′ : C′ → C′ ⊂ X ′ is an immersion by Lemma 1. Let f : C′ →C be the induced morphism on the normalizations. From the exact sequence

0 −→ On−1

C−→ µ∗T ∗(X) −→ T ∗(C) −→ 0,

we have n − 1 pointwise independent sections ω1, . . . , ωn−1 of µ∗T ∗(X) an-nihilating T (C). Furthermore, we can assume that any linear combinationof ω1, . . . , ωn−1 is non-zero on Tb(B) for any b ∈ C ∩ B. By pulling themback via f , we get n− 1 sections ω′1, . . . , ω

′n−1 of µ′∗T ∗(C′) on C′ which are

pointwise linearly independent at points different from µ′−1(R). From thegenericity of C, we may assume that f |R is unramified at C′ ∩ R. Thus forany b′ ∈ C′ ∩ R and b = f(b′) ∈ C ∩ B, the differential df : Tb′(R) → Tb(B)is an isomorphism. It follows that any linear combination of ω′1, . . . , ω′n−1

is non-zero on Tb′(R). This implies that ω′1, . . . , ω′n−1 give n − 1 pointwiseindependent sections of µ′∗T ∗(X ′) annihilating T (C′), showing that C′ hastrivial normal bundle.

Remark. An important case where Proposition 6 will be applied is whenboth X and X ′ are Fano manifolds (Section 6). Note that any curve with triv-ial normal bundle on a Fano manifold must be a rational curve and immersedrational curves through generic points have semi-positive normal bundle. Inthis case, the proof of Proposition 6 gets slightly simpler. In fact, since theconormal bundle of C is trivial, the conormal bundle of C′, which is immersedby Lemma 1, has non-negative degree because it is generated by global sec-tions at generic points by pulling back the global sections of the conormalbundle of C. It follows that K−1

C′ ≥ K−1X′ |C′ > 0 since X ′ is Fano. Thus

C′ is a rational curve. From the genericity of C, the normal bundle of C′ issemi-positive. Since the conormal bundle is generated by global sections at ageneric point, C′ has trivial normal bundle.

From Proposition 6, the components of f−1(C) for various choices of [C] ∈M define a webM′ on X ′. This webM′ is called the inverse image web ofM under f . Let ψ′ : F ′ →M′ and φ′ : F ′ → X ′ be the associated universalfamily morphisms. As before, we assume the normality of F ′ and M′. LetΦ′ : F ′ → PT (X ′) be the tangent map and C′ ⊂ PT (X ′) be the varietyof M′-tangents. Then df(C′), the strict image of C′ under the rational mapdf : PT (X ′)→ PT (X), is exactly C. The union of the components of df−1(C)which are dominant over X ′ is exactly C′. The degree of M′ is the same asthat of M.

Proposition 7. Let f : X ′ → X be a generically finite morphism betweentwo projective manifolds. Suppose there exists a web M on X and M′ is theinverse image web on X ′. Then f−1(EM) ⊂ EM′ .


Proof. Let L be an irreducible component of EM. Suppose for a genericpoint y of L, there exists y′ ∈ f−1(y) with y′ 6∈ EM′ . Let d be the degreeof M and M′. Since π′ : C′ → X ′ is unramified at y′, there exist d distinctgerms C′1, . . . , C′d of smooth curves through y′ with distinct tangent vectorsat y′ which are germs of members of M′.

If y′ is not in R, C1 := f(C′1), . . . , Cd := f(C′d) give d distinct germs ofsmooth curves through y with distinct tangents which are germs of membersof M, a contradiction to y ∈ EM.

Suppose y′ ∈ R. From the genericity of y, this means that the componentof R containing y is sent to a component of EM by f . We may assume thaty′ is a smooth point of R and dfy′ : Ty′(R) → Ty(L) is an isomorphism. Weclaim that there is at most one germ among C′1, . . . , C

′d which is not contained

in R. In fact, suppose C′1 and C′2 do not lie on R. Then f(C′1) and f(C′2) aregerms of members of M which do not lie on L. From Proposition 1, f(C′1)and f(C′2) are transversal to L at y. This implies that C′1 and C′2 are tangentto the kernel of df at y′; however, they have distinct tangent vectors at y′, acontradiction.

Suppose all of C′1, . . . , C′d are contained in R. Since dfy′ : Ty′(R)→ Ty(L)is an isomorphism, f(C′1), . . . , f(C′d) give d distinct germs of smooth curvesthrough y with distinct tangents which are germs of members of M. A con-tradiction to y ∈ L.

Suppose C′1 is not contained in R and all of C′2, . . . , C′d are contained in R.

Then f(C′1) must be transversal to L by Proposition 1, while f(C′2), . . . , f(C′d)give d − 1 distinct germs of smooth curves through y with distinct tangentvectors at y which are germs of members of M, a contradiction to y ∈ L

again.

Proposition 8. In the situation of Proposition 7, let L be a component ofEM and L′ be a component of f−1(L) ⊂ EM′ . Then τL = τL′ .

Proof. Let y ∈ L be a generic point and y′ ∈ f−1(y) ⊂ L′. Let U bean isolating neighborhood of y with respect to π : C → X and U ′ be anisolating neighborhood of y′ with respect to π′ : C′ → X ′. We may assumethat U = f(U ′). Let G (resp. G′) be the union of components of C|U (resp.C′|U ′ ) which intersect PT (EM) (resp. PT (EM′)). Recall that τL is the degreeof π on G and τL′ is the degree of π′ on G′.

The strict image of C′|U ′ under df |U ′ is contained in C|U . Since dfu′ :PTu′(X ′)→ PTu(X) is an isomorphism for a generic u′ ∈ U ′ with u = f(u′),the degree of π on the strict image df(C′|U ′) is d, the degree of M and M′.It follows that df(C′|U ′) = C|U .


The genericity of y′ implies that PTy(L′) ∩Kerdf = ∅. Thus the rationalmap df : PT (U ′)→ PT (U) is a well-defined morphism on G′. Then it is clearthat df(G′) ⊂ G and τL′ ≤ τL.

Let A (resp. A′) be the union of components of C|U (resp. C′|U ′) whichare disjoint from PT (L) (resp. PT (L′)) so that

C|U = G ∪A,C′|U ′ = G′ ∪ A′.

From Proposition 3, A (resp. A′) is disjoint from Φ(R) (resp. Φ′(R′)) whereR (resp. R′) is the ramification divisor of φ : F → X (resp. φ′ : F ′ → X ′). Byshrinking U (resp. U ′), one can assume that φ (resp. φ′) is locally one-to-oneover U (resp. U ′) outside R (resp. R′) and Φ (resp. Φ′) is a normalizationmap over U (resp. U ′) by Proposition 2. Thus π (resp. π′) is 1-to-1 on eachcomponent of A (resp. A′).

We claim that the strict image of A′ under df is contained in A. This isobvious if y′ 6∈ R, the ramification divisor of f . Thus we may assume thatL′ ⊂ R. Since π′ is 1-to-1 on each component A′ of A′, we may assume that A′

defines a foliation by smooth curves on U ′. Since A′ is disjoint from PT (L′),the leaves of this foliation are transversal to L′. Each leaf of this foliationis part of a member of M′. Thus from the genericity of y′, we may assumethat the leaves are parts of generic members ofM′ and their images under fare parts of generic members ofM. From Proposition 1, the images of leavesof the foliation on U ′ are sent to germs of curves transversal to L. Thus byshrinking U ′ if necessary, we may assume that the image of this foliation underf defines a foliation by smooth curves in a neighborhood of y whose leaves aretransversal to L. This foliation defines the strict image df(A′) which containsa unique point in C ∩ π−1(y) corresponding to the tangent vector of the leafthrough y. Since the leaf is transversal to L, df(A′) is disjoint from PT (L).Thus df(A′) is an irreducible component of A.

Let e (resp. e′) be the degree of π (resp. π′) on A (resp. A′). Since dfu′ fora generic u′ ∈ U ′ is an isomorphism, df(A′) ⊂ A implies that e ≥ e′. Sinced = τL + e = τL′ + e′, we see that τL′ ≥ τL and we are done.

Remark. One cannot rule out the possibility that an irreducible com-ponent of C|U is pulled back to a reducible set in C′|U . This point causessome complications in the statements and proofs of some results of this pa-per. For example, we are not able to extend Proposition 7 to a statement likef−1(DM) ⊂ DM′ .


4. Discriminantal orders of inverse image webs

We will continue to use the notation in the previous section. We are givena generically finite morphism f : X ′ → X between two projective manifoldsof dimension n, a web M on X and the inverse image web M′. Let M bea component of DM and L′ be a component of f−1(M) ⊂ EM′ given byProposition 7. By Proposition 4 and Proposition 8, 1 < τM = τL′ . Thus thediscriminantal orders δM and δL are well-defined. If L′ is not contained inthe ramification divisor R of f , then δL′ = δM . When L′ is a component ofR, we have the following result.

Proposition 9. Given a generically finite morphism f : X ′ → X, a webM and its inverse image M′ as above, let L′ be a component of EM′ which isat the same time a component of the ramification divisor R of f and satisfiesM := f(L′) ⊂ DM. Write r > 1 for the local sheeting number of f at ageneric point of L′ and let m > 1 be the tangential sheeting number of L′ andM . Then r ≤ mδL′ .

Proof. We may assume that r > m. Let y′ ∈ L′ be a generic point andset y = f(y′) ∈ M . We can choose a coordinate system (w1, . . . , wn) on anisolating neighborhood U ′ of y′ with respect to C′ and a coordinate system(z1, . . . , zn) on an isolating neighborhood U = f(U ′) of y with respect to C,so that f is given by

z1 = w1, . . . , zn−1 = wn−1, zn = wrn.

Note that zn = 0 defines M ∩ U and wn = 0 defines L′ ∩ U ′. Let G ⊂ C|U bethe union of components intersecting PT (M) and let G′ ⊂ C′|U ′ be the unionof components intersecting PT (L′). By definition, the tangential sheetingnumber m is the degree of π′ on G′, which is equal to the degree of π on G,by Proposition 8. Let ξ be the tautological line bundle of PT (X) restrictedto the normalization G of G. By shrinking U , we assume that ξ is trivial on Gas in Section 2. Let

V = v1∂

∂z1+ · · ·+ vn

∂

∂zn

be a section of ξ on G as in Section 2. Since G∩π−1(y) ⊂ PTy(M), vn vanishesover zn = 0 and one of v1, . . . , vn−1 is non-vanishing at points over U ∩M .Consider

f∗V := df∗v1∂

∂w1+ · · ·+ df∗vn−1

∂

∂wn−1+

df∗vn

r · wr−1n

∂

∂wn,

where df∗vi is the pull-back of the holomorphic function vi on G by the holo-morphic map df : G′ → G which is the lifting of df : G′ → G to the normaliza-tions. f∗V is a meromorphic section of the tautological bundle ξ′ of PT (X ′)over G′.


We claim that the meromorphic function df∗vnwr−1n

on G′ must be holomorphic.Suppose not. Since we are interested in a generic point y′ ∈ L′, we may assumethat G′ is smooth and a generic point a of the polar divisor of df∗vn

wr−1n

lies over

G′∩PT (L′). Let h be a holomorphic function on G′ defining the polar divisorso that hdf

∗vnwr−1n

is non-vanishing at a. Thus hf∗V is a non-vanishing section

of ξ′ in a neighborhood of a in G′. However, at a point of h = 0,

h f∗V = hdf∗vn

wr−1n

∂

∂wn

corresponds to a tangent vector transversal to L′. This is a contradiction tothe fact that G′ ∩ π−1(y′) ⊂ PTy′(L′).

Since one of df∗v1, . . . , df∗vn−1 is non-vanishing on G′, we see that f∗V is

a non-vanishing section of ξ′ over G′. Thus we can compute the discriminantalorder δL′ using f∗V . As in Section 2, for a given Q = (qij) ∈ o(n,C), let

ΓQ(x) =∏

1≤α6=β≤m

n∑i,j=1

qijvi(xα)vj(xβ)

,

Γ′Q(x′) =∏

1≤α6=β≤m

n∑i,j=1

qijv′i(uα)v′j(u

β)

,

where u1, . . . , um are the points of G′ over a generic point u ∈ U ′,x1 = dfu(u1), . . . , xm = dfu(um) are the points of G over x = f(u), and

v′1 = df∗v1, . . . , v′n−1 = df∗vn−1, v′n =

df∗vn

r · wr−1n

.

For each µ = (α, β) ∈ I, we define

Λµ :=n−1∑i,j=1

qijvi(xα) · vj(xβ),

Λ′µ :=n−1∑i,j=1

qijdf∗vi(uα) · df∗vj(uβ),

Ωµ :=n−1∑i=1

qinvi(xα)vn(xβ) +n−1∑i=1

qnivi(xβ)vn(xα),

Ω′µ :=n−1∑i=1

qindf∗vi(uα)df∗vn(uβ)r · wr−1

n

+n−1∑i=1

qnidf∗vi(uβ)df∗vn(uα)r · wr−1

n

,

so that

Γ′Q(u) =∏µ∈I

(Λ′µ + Ω′µ).


To prove Proposition 9, it suffices to show that the vanishing order of Γ′Q onwn = 0 is ≥ r

m .Consider the holomorphic arc ρ(t) ∈ U, t ∈ ∆ε defined by z1 = · · · =

zn−1 = 0, zn = t, and the holomorphic arc ρ′(s) ∈ U ′, s ∈ ∆ε′ defined byw1 = · · · = wn−1 = 0, wn = s. f restricted to these arcs is given by t = sr. Byrestricting to these holomorphic arcs, we regard Λµ and Ωµ as multi-valuedfunctions of the variable t and Λ′µ and Ω′µ as multi-valued functions of thevariable s. By Puiseux expansion, we can consider the fractional vanishingorders λµ (resp. ωµ) of Λµ (resp. Ωµ) in t and the fractional vanishing ordersλ′µ (resp. ω′µ) of Λ′µ (resp. Ω′µ) in s. Since

Λ′µ = df∗Λµ,

Ω′µ =1

r · wr−1n

df∗Ωµ,

we have λ′µ = rλµ and ω′µ = rωµ − (r − 1) = (ωµ − 1)r + 1.From y ∈ M , there exists an irreducible component of Go of G which is

generically k-to-1 over π(Go) for some k > 1. Regard V as a non-vanishingsection of ξ over Go and vi as holomorphic functions on the normalization Go.By Puiseux expansion, there exist convergent power series g1(τ), . . . , gn(τ) ofa variable τ , so that

v1|ρ(∆ε) = g1(t1k ), . . . , vn|ρ(∆ε) = gn(t

1k ).

Let g be the order of gn(τ) in τ . Then

df∗vn

r · wr−1n

|ρ′(∆ε′ )=

gn(srk )

r · sr−1

has vanishing order grk − r + 1 in s, which must be non-negative. So we get

g ≥ k(r−1)r . From the assumption r > m, we see g ≥ k.

Choose ν = (α, β) ∈ I so that both xα and xβ lie on Go. Then both λν

and ων are ≥ 1k . For ζ = e

2π√−1k , we can write

Ων |ρ(∆ε) =n−1∑i=1

qin(gi(ζαt

1k ) · gn(ζβt

1k )− gi(ζβt

1k ) · gn(ζαt

1k )).

We claim that this has vanishing order ≥ k + 1 in t1k . The claim is obvious

if g > k. So assume that g = k. Suppose gi has no constant term. Then itis obvious that the i-th term of Ων has vanishing order ≥ g + 1 = k + 1 int

1k . Suppose gi has a non-zero constant term. Then the lowest order term

of gi(ζαt1k ) · gn(ζβt

1k ) cancels that of gi(ζβt

1k ) · gn(ζαt

1k ) because ζg = 1.

Thus in all cases, the vanishing order of Ων is ≥ k + 1 in t1k . In other words,


ων ≥ 1 + 1k . We conclude that λ′ν = rλν ≥ r

k and ω′ν = (ων − 1)r+ 1 ≥ rk + 1.

Thus the vanishing order of

Γ′Q(u) =∏µ∈I

(Λ′µ + Ω′µ)

is ≥ min(λ′ν , ω′ν) ≥ min( rk ,rk + 1) ≥ r

m .

5. Deformation rigidity of generically finite morphisms

Let X be a projective manifold and C be a rational curve with trivial nor-mal bundle on X . Since H1(P1,On−1) = 0, we can deform C as rationalcurves to cover an open subset of X . Thus the existence of a rational curvewith trivial normal bundle is equivalent to the existence of a web of rationalcurves. As mentioned in the introduction, this is also equivalent to the ex-istence of a family of rational curves covering X which have degree 2 withrespect to K−1

X . This property is preserved under deformation:

Proposition 10. Let χ : X → ∆ be a regular family of projective mani-folds. Suppose X0 := χ−1(0) has a web M0 of rational curves; then, for someε > 0, Xt := χ−1(t) has a web Mt of rational curves for all 0 < |t| < ε sothat Mt forms a flat family over 0 < |t| < ε and members of M0 are limitsof members of Mt.

In the statement of Proposition 10, we do not exclude the possibility thatthe limit ofMt at t = 0 has other components different from M0.

Proof. Let C be a generic member of a component of M0. Since C hastrivial normal bundle in X0, it has trivial normal bundle in the complex mani-fold X . Thus C can be deformed as rational curves to cover an open subset ofX . This also follows from Kodaira stability ([Kd]) since H1(P1, NC⊂X0) = 0for the normal bundle NC⊂X0

∼= On−1C . Let D be the irreducible component

of the normalized Douady-Hilbert scheme of rational curves in X containingthe point [C] corresponding to C. Then D is smooth at [C] and the naturalprojectionD → ∆ is of maximal rank at [C]. Let Dt be the the fiber of D → ∆at t ∈ ∆. Since all members of Dt have degree 2 with respect to K−1

Xt, a com-

ponent of Dt whose members cover Xt is an irreducible web on Xt. Let Dotbe the union of the components of Dt whose members cover Xt. Then letMt

be the union of all Dot which arises as we choose C from different componentsof M0. Clearly, Mt, 0 < |t| < ε form a flat family for a suitable choiceof ε.


Proposition 11. Let χ : X → ∆ be as in Proposition 10 and Ct ⊂ PT (Xt)be the variety of Mt-tangents. For a given projective manifold X ′, let X ′ :=X ′×∆. Suppose there exists a morphism f : X ′ → X respecting the projectionsto ∆, such that ft : X ′ → Xt is generically finite for each t ∈ ∆. Then thereexists a subvariety C′ ⊂ PT (X ′) so that the union of components of df−1

t (Ct)dominant over X ′ is C′ whenever 0 < |t| < ε; the union of components ofdf−1

0 (C0) dominant over X ′ is contained in C′.Proof. Let C′t ⊂ PT (X ′) be the union of components of df−1

t (Ct) dominantover X ′. C′t is a flat family for 0 < |t| < ε. C′0 is contained in the closure ofthe union of C′t, 0 < |t| < ε. From Section 3, C′t, 0 < |t| < ε gives a flatfamily of webs on X ′. Since there are only countably many webs on X ′, C′tmust be independent of t. Just set C′ = C′t for any t, 0 < |t| < ε.

Note that any web on a Fano manifold is a web of rational curves. In theprevious sections, we made the assumption that the discriminantal divisor ofa web is non-empty. This is always the case for webs on a Fano manifold ofPicard number 1:

Proposition 12. Let X be a Fano manifold of Picard number 1 andM be aweb. Let ψ : F →M and φ : F → X be the universal family morphisms. Thenφ is not birational on any component of F . Consequently, the discriminantaldivisor DM ⊂ X is non-empty.

Proof. We may assume that M and F are irreducible. Recall that φ isunramified in an analytic neighborhood of a generic fiber ψ−1(v), v ∈ M.Choose a hypersurface H ⊂ M disjoint from v such that φ(ψ−1(H)) is ahypersurface in X . If φ is birational, the curve φ(ψ−1(v)) is disjoint from thehypersurface φ(ψ−1(H)) in X , a contradiction to the fact that X is of Picardnumber 1.

Now let us go to the situation of Theorem 1:Proposition 13. Let χ : X → ∆ := t ∈ C, |t| < 1 be a regular family

of Fano manifolds of Picard number 1 so that X0 = χ−1(0) has a web M0 ofrational curves. By Proposition 10, we have a web Mt on Xt for 0 < |t| <ε. Let Zt ⊂ Xt be the subvariety of codimension > 1 so that the universalfamily morphism φt : Ft → Xt associated to the web Mt is unramified overXt −DMt − Zt. Given f : X ′ → X as in Proposition 11, let M′ be the webdefined by C′ on X ′. For a generic member C ⊂ X0 ofM0 and any irreduciblecomponent C′ of f−1

0 (C), the following holds:

(i) C′ intersects EM′ transversally;(ii) Ct := ft(C′) is a member of Mt having trivial normal bundle, is not

contained in DMt and is disjoint from Zt for all 0 < |t| < ε.Proof. This is a direct consequence of Proposition 1 and DMt ⊂ ft(EM′)

which follows from Proposition 7.


Proposition 14. Under the assumptions of Proposition 13, for each fixedt ∈ ∆, the intersection Ct ∩DMt contains at least two distinct points on thenormalization Ct of Ct.

Proof. Since Xt has Picard number 1, Ct has non-empty intersection withDMt . Suppose the intersection is only at one point on Ct. Then the normal-ization of Ct − DMt is biholomorphic to C and is simply connected. SinceFt is unramified over Xt − DMt − Zt, this means that φ−1

t (Ct) consists ofdt distinct irreducible components, where dt is the degree of φt : Ft → Xt.This is a contradiction to Proposition 12 and the next lemma applied to eachirreducible component W of Ft.

Lemma 2. Let η : W → X be a generically finite morphism from anirreducible normal variety W onto a Fano manifold X with Picard number 1which has a web M. Suppose for a generic member C of M, each componentof the inverse image η−1(C) is birational to C by η. Then η : W → X itselfis birational.

Proof. We may assume that W is a projective manifold by desingulariza-tion. Suppose η is not birational. Let B ⊂ X be the branch divisor of ηand R ⊂ W be the ramification divisor of η. By the assumption on η, eachcomponent C1 of η−1(C) is a rational curve. By Proposition 6, C1 has trivialnormal bundle. So we have KW · C1 = −2 = KX · C = η∗KX · C1. Thisimplies C1 is disjoint from the ramification divisor R ⊂ W . Since this holdsfor any component C1 of η−1(C), C is disjoint from B, a contradiction to theassumption that X is of Picard number 1.

Proposition 15. Fix a complex number s, 0 < |s| < ε. Under the as-sumptions of Proposition 14, for a generic choice of C ⊂ X0 and an irre-ducible component C′ of f−1

0 (C), let y1, y2 ∈ C′ be any two points satisfyingfs(y1) = fs(y2). Then ft(y1) = ft(y2) whenever |t| < ε.

Proof. By continuity, it suffices to prove ft(y1) = ft(y2) for t satisfying|t− s| < δ for some positive real number δ < min(|s|, ε − |s|). By our choiceof C, Ct = ft(C′) is a member of Mt with trivial normal bundle in Xt.Since Xt is Fano, Ct is a rational curve. From Proposition 1 and Proposition13, we can choose a generic C and ε > 0, so that Ct = ft(C′) intersects DMt

transversally for all 0 < |t| < ε. By Proposition 14, there exists a simultaneousnormalization µt : P1 → Ct for all t satisfying 0 < |t − s| < δ so that µt(o)and µt(∞) lie in Ct ∩DMt for two fixed distinct points o,∞ in P1. Also,we can assume that C′ is transversal to EM′ .

Let ft : C′ → P1 be the pull-back of ft|C′ to the normalizations.ft, |t − s| < δ gives a holomorphic family of meromorphic functions onC′ with zeroes and poles lying over the points of C′ intersecting f−1

t (µt(o))and f−1

t (µt(∞)). Since f−1t (DMt) ⊂ EM′ from Proposition 7, these points


are contained in the finite set C′ ∩ EM′ which does not depend on t. Sothis family of meromorphic functions must have the same zeroes and poles onC′. Thus ft = h(t)f0 for some non-vanishing holomorphic function h(t) on|t− s| < δ, from which the assertion is obvious.

We are ready to prove:

Theorem 1. Let χ : X → ∆ := t ∈ C, |t| < 1 be a regular family ofFano manifolds of Picard number 1 so that X0 = χ−1(0) has rational curveswith trivial normal bundles. For a given projective manifold X ′, suppose thereexists a surjective morphism f : X ′ = X ′ ×∆→ X respecting the projectionsto ∆ so that ft : X ′ → Xt is a generically finite morphism for each t ∈ ∆.Then there exists ε > 0 and a holomorphic family of biholomorphic morphismsvt : Xo → Xt, |t| < ε, satisfying ft = vt fo, v0 = id.

Proof. By assumption, X0 has a web M0. We will use the notation fromPropositions 13–15.

Suppose that f is birational. Then f is biholomorphic over X −Z where Zis a subvariety of codimension ≥ 2. On X ′ we have the vector field V lifting d

dt

on ∆. f∗V is a vector field on X −Z. By Hartogs, we can extend it to a vectorfield on X which generates the required family of biholomorphic morphisms.

Suppose that ft is not birational, but generically e-to-1. We will constructa new projective manifold X, a generically finite dominant rational map ν :X ′ → X and a holomorphic family of generically finite morphisms gt : X →Xt, so that ν is not birational and ft = gt ν over generic points of X ′. Thisproves Theorem 1 by induction on e.

Fix s as in Proposition 15. We say that a reduced 0-cycle y1 + · · ·+ yk oflength k on X ′ is special if fs(y1) = · · · = fs(yk) and we can find a sequenceof points z1, . . . , zm so that z1, . . . , zm = y1, . . . , yk as subsets of X ′ andthere exist irreducible curves l1, . . . , lm−1 which are members of the inverseimage web of M0 so that zi, zi+1 ∈ li for 1 ≤ i ≤ m − 1. All special cycleshave length ≤ e and there are special cycles of length > 1 containing a genericpoint of X ′, otherwise ft is birational by Lemma 2.

The set of all special cycles on X ′ gives a constructible subset of the Hilbertscheme of 0-dimensional subschemes of X ′. We can find an irreducible com-ponent of this set so that the corresponding cycles cover an open set of X ′.Make such a choice with maximal possible length. Let S be the closure ofthat component and let σ : A → S and λ : A → X ′ be the universal familymorphisms so that σ is flat, but not birational. We claim that λ is birational.


Otherwise we have two distinct special cycles containing a given generic pointy of X ′, from which we can construct a special cycle of strictly longer lengthcontaining y, a contradiction to the choice of S.

Let X be a desingularization of S and σ : A → X, λ : A → X ′ be theinduced morphisms. Define the rational map ν : X ′ → X as ν := σ λ−1.Then ν is a generically finite dominant rational map which is not birational.Consider the morphism ft = ft λ from A to Xt. From Proposition 15, ageneric fiber of σ is contained in a fiber of ft. Thus each fiber of σ is containedin a fiber of ft by the flatness of σ. It follows that we get a morphismgt : X → Xt satisfying ft = gt ν. Since X ′, Xt are all projective, it is easyto see that gt is a holomorphic family.

An immediate consequence of Theorem 1 is:

Corollary 1. For any irreducible complete variety X ′ of dimension ≥ 3,there are only countably many Fano manifolds of Picard number 1 having ra-tional curves with trivial normal bundles, which can be the image of a gener-ically finite morphism from X ′.

In [HM1], we showed that for any Fano manifold S of Picard number 1which is homogeneous, a finite morphism f : S → X onto a projective mani-fold X must be a biholomorphism unless X ∼= Pn. It is natural to ask whathappens if S is almost homogeneous. We will consider a special case here.A Mukai-Umemura threefold (cf. [MU]) is a Fano threefold of Picard num-ber 1 which is almost homogeneous under PGL2. There are two such Fanothreefolds, one having the octahedral group as the isotropy group, and theother having the icosahedral group as the isotropy group. Note that both theoctahedral group and the icosahedral group are maximal finite subgroups inPGL2 (e.g. [YY]).

Corollary 2. Let S be a Mukai-Umemura threefold and X be a smoothvariety of positive dimension. If f : S → X is a surjective morphism, theneither X ∼= P3 or f is biholomorphic.

Proof. Since S is of Picard number 1, f is a finite morphism and X isa Fano threefold. If X is different from P3, it is either the 3-dimensionalhyperquadric or a Fano threefold having a rational curve with trivial normalbundle. The PGL2-action on S descends to X by [HM2] for the hyperquadricand by Theorem 1 for the others. Thus X is almost homogeneous underPGL2, and the isotropy subgroup is a finite subgroup of PGL2 containingthe isotropy subgroup for S. However, the isotropy subgroup for S, either theoctahedral group or the icosahedral group, is a maximal finite subgroup ofPGL2, a contradiction unless f is 1-to-1.


6. Boundedness of degrees of finite morphisms

In this section we will prove:

Theorem 2. Let X and X ′ be n-dimensional Fano manifolds of Picardnumber 1. Assume that X has rational curves with trivial normal bundles.Then there exists a positive number N determined by X ′ such that for anyfinite morphism f : X ′ → X, the degree of f is bounded by N .

In fact, N can be calculated from the degrees of extended discriminan-tal divisors and discriminantal orders of webs on X ′, together with Chernnumbers of X ′ and X . Note that there are only finitely many webs on X ′

and X :

Proposition 16. On a Fano manifold, there are only finitely many webs.

Proof. Any curve with trivial normal bundle on a Fano manifold must bea rational curve and its degree with respect to the anti-canonical bundle is 2.Since the anti-canonical bundle is ample, we get the finiteness of irreduciblewebs. Since a web is just the finite union of irreducible webs, there are onlyfinitely many webs.

Proposition 17. Let X ′ be an n-dimensional Fano manifold. Then thereexists a positive number N1 determined by X ′ with the following property: forany finite morphism f : X ′ → X onto a Fano manifold X which has a webM and for any component L′ of f−1(DM), the local sheeting number of f ata generic point of L′ is bounded by N1.

Proof. The local sheeting number is bounded by a number determinedby the discriminantal orders of L′ from Proposition 9. Since there are onlyfinitely many possibilities of M′ from Proposition 16, there are only finitelymany possibilities for L′. Thus we can find a bound N1.

Proof of Theorem 2. Choose a webM on X and let C be a generic memberofM. C is an immersed rational curve. For a given finite morphism f : X ′ →X , let M′ be the inverse image web of M. Let C′ be a member of M′so that f(C′) = C. Let h : C′ → C be the the pull-back of f |C to thenormalization.

We may assume that C (resp. C′) intersects DM (resp. EM) transversally.For z ∈ C′, let ez be the ramification index of h at z. If rz is the local sheetingnumber of f at µ(z) where µ : C′ → X ′ is the normalization onto its image,


then ez = rz − 1. For x ∈ C, let |h−1(x)| be the number of points of C′ overx. When m is the degree of h,∑

z∈C′∩f−1(DM)

ez =∑

x∈C∩DM

(m− |h−1(x)|)

= m C ·DM −∑

x∈C∩DM

|h−1(x)|

= m C ·DM − C′ · f−1(DM)

≥ 2m− C′ · EM′

where we used f−1(DM) ⊂ EM′ from Proposition 7 and C · DM ≥ 2 fromProposition 14. The readers can easily see below that for the proof of Theorem2, the obvious inequality C ·DM ≥ 1 is, in fact, sufficient. From Proposition17, ∑


ez ≤∑


(N1 − 1)

≤ (N1 − 1) C′ ·EM′ .It follows that m ≤ N1

2 C′ ·EM′ .From Proposition 16, there are only finitely many webs on X ′. Consider

the union L of extended discriminantal divisors of all the webs on X ′. LetL = N2K

−1X′ in Pic(X ′) ⊗ Q for some positive rational number N2. Then

C′ ·EM′ ≤ C′ · L ≤ 2N2. Thus m ≤ N1N2.Let R ⊂ X ′ be the ramification divisor R of f with the multiplicity given

by the ramification indices of each component. Then R = f∗K−1X −K

−1X′ and

R · C′ = 2m− 2. It follows that R = (m− 1)K−1X′ and f∗K−1

X = K−1X′ + R =

mK−1X′ in Pic(X ′). Thus

deg(f) =(f∗K−1

X )n

(K−1X )n

= mn (K−1X′ )

n

(K−1X )n

≤ (N1N2)n(K−1X′ )

n.

As an immediate consequence, we obtain:Corollary 3. A non-constant self-morphism f : X → X of a Fano mani-

fold of Picard number 1 which has rational curves with trivial normal bundlesis an isomorphism.

Proof. If f has degree d > 1, the m-th successive composition fm : X → X

gives a finite morphism of degree dm, where m can be arbitrary large; acontradiction to Theorem 2.

Now we obtain the following partial answer to the Question discussed inthe introduction of this paper.

Corollary 4. Let X ′ be a Fano manifold of Picard number 1. Amid Fanomanifolds of Picard number 1 which have rational curves with trivial normal


bundles, only finitely many can be the image of a holomorphic map from X ′.Furthermore, for each such X there are at most a finite number of non-trivialholomorphic maps f : X ′ → X up to automorphisms of X.

Proof. Suppose there exists an infinite sequence fi : X ′ → Xi of holomor-phic maps to Fano manifoldsXi of Picard number 1 which have rational curveswith trivial normal bundles. By the boundedness of Fano manifolds of a givendimension ([Kl]), we may assume that Xi’s are members of an irreducible flatfamily of Fano manifolds. By a pluri-anti-canonical embedding, we can re-gard Xi’s as submanifolds in some projective space PN with the same Hilbertpolynomial. The graphs of fi’s are Fano submanifolds in X ′ × PN , whichare of bounded degree with respect to K−1

X′ ⊗ O(1) from Theorem 2, whereO(1) denotes the hyperplane bundle on PN . Thus there are finitely manyirreducible families of subvarieties in X ′ ×PN , so that the graphs of fi’s aredense in each family. Generic members of such a family must be isomorphic toX ′ under the projection X ′ ×PN → X ′. Thus they give a continuous familyof finite morphisms from X ′ to a family of Fano manifolds having rationalcurves with trivial normal bundles and must be related by automorphismsof the images by Theorem 1. It follows that there are only finitely manyisomorphism classes of fi’s up to automorphisms of the images.

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Memoirs AMS 105 (1993).

Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Seoul 130-

012, Korea

E-mail address: [email protected].

Department of Mathematics, The University of Hong Kong, Pokfulam Road,

Hong Kong

E-mail address: [email protected]

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